Estimating the Pair of Factor Numbers via Eigenvalue Ratios or Rank Minimization.
Description
The function is to estimate the pair of factor numbers via eigenvalue-ratio corresponding to RMFA method or rank minimization and eigenvalue-ratio corresponding to Iterative Huber Regression (IHR).
Usage
KMHFA(X, W1 = NULL, W2 = NULL, kmax, method, max_iter = 100, c = 1e-04, ep = 1e-04)
Arguments
X
Input an array with T×p1×p2, where T is the sample size, p1 is the the row dimension of each matrix observation and p2 is the the column dimension of each matrix observation.
W1
Only if method="E_RM" or method="E_ER", the inital value of row loadings matrix. The default is NULL, which is randomly chosen and all entries from a standard normal distribution.
W2
Only if method="E_RM" or method="E_ER", the inital value of column loadings matrix. The default is NULL, which is randomly chosen and all entries from a standard normal distribution.
kmax
The user-supplied maximum factor numbers. Here it means the upper bound of the number of row factors and column factors.
method
Character string, specifying the type of the estimation method to be used.
"P",
the robust iterative eigenvalue-ratio based on RMFA
"E_RM",
the rank-minimization based on IHR
"E_ER",
the eigenvalue-ratio based on IHR
max_iter
Only if method="E_RM" or method="E_ER", the maximum number of iterations in the iterative Huber regression algorithm. The default is 100.
c
A constant to avoid vanishing denominators. The default is 10−4.
ep
Only if method="E_RM" or method="E_ER", the stopping critetion parameter in the iterative Huber regression algorithm. The default is 10−4×Tp1p2.
Details
If method="P", the number of factors k1 and k2 are estimated by
where kmax is a predetermined value larger than k1 and k2. λj(⋅) is the j-th largest eigenvalue of a nonnegative definitive matrix. See the function MHFA for the definition of Mcw and Mrw. For details, see He et al. (2023).
Define D=min(Tp1,Tp2,p1p2),
Σ^1=T1∑t=1TF^tF^t⊤,Σ^2=T1∑t=1TF^t⊤F^t,
where F^t,t=1,…,T is estimated by IHR under the number of factor is kmax.
If method="E_RM", the number of factors k1 and k2 are estimated by
He, Y., Kong, X., Yu, L., Zhang, X., & Zhao, C. (2023). Matrix factor analysis: From least squares to iterative projection. Journal of Business & Economic Statistics, 1-26.
He, Y., Kong, X. B., Liu, D., & Zhao, R. (2023). Robust Statistical Inference for Large-dimensional Matrix-valued Time Series via Iterative Huber Regression. <arXiv:2306.03317>.